Is 497 a Prime Number?

There are two types of numbers, mostly —

Prime numbers and composite numbers, depending on the number of factors.

A prime number is a natural number that is divisible only by 1 and itself.

For example, 3 is a prime number because it is divisible by 1 and itself.

A composite number is a positive number that is divisible by more than two numbers.

For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers follow a few properties like:

• Prime numbers are positive numbers always greater than 1.
   
• 2 is the only even prime number.
   
• They have only two factors: 1 and the number itself.
   
• Any two distinct prime numbers are co-prime numbers because they have only one common factor, which is 1.

As 497 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 497 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. A few methods are:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization

The method in which we count the number of divisors to categorize the numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize prime and composite numbers. If there is a total count of only 2 divisors, then the number would be prime. If the count is more than 2, then the number is composite. Let’s check whether 497 is prime or composite.

Step 1: All numbers are divisible by 1 and itself.

Step 2: Divide 497 by 2. It is not divisible by 2, so 2 is not a factor of 497.

Step 3: Divide 497 by 3. It is not divisible by 3, so 3 is not a factor of 497.

Step 4: You can simplify checking divisors up to 497 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 497 by 7, it is divisible by 7.

Since 497 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is not even, so 497 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 497 is 20. Since 20 is not divisible by 3, 497 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is not 0 or 5, so 497 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (7 × 2 = 14). Then, subtract it from the rest of the number (49 - 14 = 35). Since 35 is divisible by 7, 497 is divisible by 7. Since 497 is divisible by 7 and other numbers, it has more than two factors.

Therefore, it is a composite number.

The prime number chart is a tool created by using a method called “The Sieve of Eratosthenes.” In this method, we follow the following steps.

Step 1: Write numbers in a large enough range to include 497.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all the multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all the multiples of 3.

Step 5: Repeat this process until you reach the number 497. Through this process, we will have a list of prime numbers.

497 is not present in the list of prime numbers, so it is a composite number.

Prime factorization is a process of breaking down a number into prime factors. Then multiply those factors to obtain the original number.

Step 1: We can write 497 as 7 × 71.

Step 2: Both 7 and 71 are prime numbers.

Step 3: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 497 is 7 × 71.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 12 is a composite number because 12 is divisible by 1, 2, 3, 4, 6, and 12.
   
• Co-prime: Two numbers are co-prime if their greatest common divisor is 1. For example, 8 and 15 are co-prime numbers.
   
• Prime numbers: A natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 17 is a prime number.
   
• Prime factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 30 is 2 × 3 × 5.
   
• Divisibility rules: Guidelines to determine whether a number is divisible by another without performing division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.